We provide a comparative study of the Subspace Projected Approximate Matrix method, abbreviated SPAM, which is a fairly recent
iterative method of computing a few eigenvalues of a Hermitian matrix A. It falls in the category of inner-outer iteration
methods and aims to reduce the costs of matrix-vector products with A within its inner iteration. This is done by choosing
an approximation A 0 of A, and then, based on both A and A 0, to define a sequence (A k ) k=0 n of matrices that increasingly
better approximate A as the process progresses. Then the matrix A k is used in the kth inner iteration instead of A.

In
spite of its main idea being refreshingly new and interesting, SPAM has not yet been studied in detail by the numerical linear
algebra community. We would like to change this by explaining the method, and to show that for certain special choices for
A 0, SPAM turns out to be mathematically equivalent to known eigenvalue methods. More sophisticated approximations A 0 turn
SPAM into a boosted version of Lanczos, whereas it can also be interpreted as an attempt to enhance a certain instance of
the preconditioned Jacobi-Davidson method.

Numerical experiments are performed that are specifically tailored to
illustrate certain aspects of SPAM and its variations. For experiments that test the practical performance of SPAM in comparison
with other methods, we refer to other sources. The main conclusion is that SPAM provides a natural transition between the
Lanczos method and one-step preconditioned Jacobi-Davidson.

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